Temperature transformation recovering the compressible law of the wall for turbulent channel flow

TL;DR

This work derives VD-type and SL-type temperature transformations for compressible turbulent channel flow by analyzing momentum and energy balances in the overlap region while explicitly accounting for mixing-length modeling, body-force work, and TKE transport. The transformations are designed to collapse the temperature profile onto its incompressible counterpart across isothermal and adiabatic walls, with the SL-type showing superior sublayer and buffer-layer performance when using suitable mixing-length models (e.g., parabolic $l_m^P$ or enhanced $l_m^E$). Key findings include extended logarithmic regions for temperature with the enhanced mixer, robust performance across DNS/WRLES data sets, and diagnostic validation via a log-law residual function. The results offer a pathway to improved near-wall modeling and potential extensions to broader compressible-flow configurations, emphasizing the need for accurate TKE-transport treatment and careful mixing-length selection. Overall, the proposed framework advances the ability to represent compressible heat-transfer behavior in the near-wall region with practical implications for LES wall models and high-Reynolds-number predictions.

Abstract

Velocity and temperature distributions are both crucial for modeling compressible wall-bounded turbulent flows. The compressible law of the wall for velocity has been extensively examined through velocity transformations. However, the issue of a well-established temperature transformation remains open. We propose a new temperature transformation for compressible turbulent channel flow. Our approach is based on the analysis of momentum and energy balance equations in the overlap layer. It accounts for the influences of mixing length model, the work of the body force, and the turbulent kinetic energy transport. Two types of temperature transformations are obtained: Van Driest type (VD-type) and semi-local type (SL-type). The performance of these transformations is evaluated using data from direct numerical simulations and wall-resolved large eddy simulations of compressible turbulent channel flow. Both the VD-type and SL-type transformations apply to isothermal and adiabatic walls. The SL-type transformation provides better data collapse in the viscous sublayer and buffer layer, thereby recovering the temperature law of the wall. When a suitable mixing length model is applied, the SL-type transformation yields results that agree with the incompressible temperature profile or exhibit extended logarithmic behavior. Results from the present study highlight careful consideration of the turbulent kinetic energy transport term in different thermal boundary conditions. Applications of the proposed transformation in near-wall modeling and its potential extension to more general configurations are also discussed.

Figures (17)

• Figure 1: Momentum (a) and energy (b) balance in a statistically steady turbulent channel flow. $h$ is the channel half-height. The black dashed line indicates a reference $y$-plane in the lower half-channel ($y/h$ = 0 to 1). Th expressions for each heat flux component are provided in Eqs. \ref{['eq:q_mu_T']} to \ref{['eq:q_f']}.
• Figure 2: Energy budget in compressible turbulent channel flow. (a, d): isothermal wall with $M_b = 1.7, Re_b = 10000$ in classical isothermal setup (see table \ref{['table:WRLES_JAXFluids']}); (b, e): isothermal wall side with $M_b = 1.86, Re_b = 20813$ in mixed isothermal/adiabatic configuration (case "iF2" in table \ref{['table:DNS_LC2022_isothermal_wall']}); (c, f): adiabatic wall side with $M_b = 1.86, Re_b = 45788$ in mixed isothermal/adiabatic configuration (case "aF2" in table \ref{['table:DNS_LC2022_adiabatic_wall']}). Here, $M_b$ and $Re_b$ represent the bulk Mach number and bulk Reynolds number, respectively, as defined in § \ref{['sec:performance_of_temperature_transformation']}. The heat flux in panel (c) is normalized using $q_w$ from the corresponding isothermal wall side. Negative values indicate heat flux away from the wall (see Fig. \ref{['fig:momentum_energy_balance']}).
• Figure 3: Distribution of mixing length model and its influence on the transformed temperature. $l_m^L$, $l_m^P$, $l_m^E$, and $l_m^{LD}$ correspond to models given by Eqs. \ref{['eq:lm_linear']}, \ref{['eq:lm_parabolic']} , \ref{['eq:lm_new']}, and \ref{['eq:lm_linear_damp']}, respectively. Cases included are: $M_b = 0.30$ and $Re^*_\tau = 177$ for (a, d, g), $M_b = 2.39$ and $Re^*_\tau = 245$ for (b, e, h), $M_b = 1.57$ and $Re^*_\tau = 965$ for (c, f, i). DNS data from Gerolymos2023Gerolymos2024aGerolymos2024b are employed. The black dotted lines represent the theoretical value from DNS using $l_m = {{(-\widetilde{u^{\prime\prime} v^{\prime\prime}})}^{1/2}}/{(d\tilde{u}/dy)}$. Points $L$ and $U$ are the approximate lower and upper bound of the logarithmic region using $l_m^P$. The black dashed lines in (g, h, i) represent the incompressible result of Pirozzoli2016a for the temperature profile in turbulent channel flow at $Re_\tau \approx 4000$ and $Pr = 0.71$.
• Figure 4: Temperature profiles above the isothermal wall under the VD-type transformation of (a) Chen2022, (b) Huang2023, and (c) the present transformation given by Eq. \ref{['eq:VD_Tplus']}, using DNS data from Gerolymos2023Gerolymos2024aGerolymos2024b. Additional details are provided in table \ref{['table:DNS_GV2024']}. All subfigures share the same color bar. In panel (c), results from $l_m^P$ and $l_m^L$ are shifted upward by 5 and 10 units, respectively. Black dashed line: the incompressible DNS result of Pirozzoli2016a for the temperature profile in turbulent channel flow at $Re_\tau \approx 4000$ and $Pr = 0.71$.
• Figure 5: Temperature profiles above the isothermal wall under the SL-type transformation of (a) Chen2022, (b) Huang2023, (c) Cheng2024b, and (d) the present transformation given by Eq. \ref{['eq:SL_Tplus']}, using DNS data from Gerolymos2023Gerolymos2024aGerolymos2024b. Additional details are provided in table \ref{['table:DNS_GV2024']}. All subfigures share the same color bar. In panel (d), results from $l_m^P$ and $l_m^L$ are shifted upward by 5 and 10 units, respectively. The black dashed lines are the same as Fig. \ref{['fig:GV2024_VD_Tplus_yplus']}. The blue dotted line corresponds to incompressible case at $Re_\tau \approx 1000$ from Pirozzoli2016a.